8 DEPENDENCE OF THE SOLUTION ON INITIAL CONDITIONS 303

(Use the mean value theorem (8.5.1 >, as well as (7.5.5).)

(c) Deduce from (b) that, for n ^ N

\zn(t)-zn(t~h)\^ f w(s9w(s) + S)ds
Jt~h

(consider in succession the cases zn(t) ^ zn(t — h) and zn(t) ^ zn(t — hj). Hence
MO-w(f-W|^ f oj(s,w(s))ds (by (8.7.8)).

Vt-Ji

(d) Conclude that w(t) = 0 in J (same argument as in Problems 4(b) and 5(a))} and
using Problem 7, prove that the sequence (un) converges uniformly in J to a solution of
x = f(t, x) taking the value x0 for / = 0.

9. The notations being those of Section 10.4, suppose E is finite dimensional, and/is
continuous and bounded in I x H. Suppose in addition there is at most one solution
of x' = f(t, x) defined in any open interval J <= I containing t0, and equal to x0 e H
for t = /0 • Suppose that, for any integer n > 0, there exists an approximate solution
un of x'=f(t,x), with approximation l/n, defined in I and taking its values in H,
and such that un(t0) = x0 . Show that in any compact interval contained in I, the sequence
(un) is uniformly convergent to a solution u of xf =/(/, x), taking its values in H and
such that w(/o) == *o. (Use the same argument as in Problem 7.)

8. DEPENDENCE OF THE SOLUTION ON INITIAL CONDITIONS

(10.8.1) Let f be locally lipschitzian (10.5.4) in I x H z/K =R, analytic in
I x H ifK = C. Then, for any point (a,b)elxH :

(a) There is an open ball J c I of center a and an open ball V c H of
center b such that, for every point (t0, ;\;0) e J x V, there exists a unique solution
t-+u(t, tQ, x0) of (10.4.1) defined in J, taking its values in H and such that

U(tQ , tQ , XQ) = XQ .

(b) The mapping (t, t0, x0) -> u(t, t0, x0) is uniformly continuous in
J x J x V.

(c) There is an open ball W c V of center b such that, for any point
(t, t0, x0) e J x J x W, the equation x0 = u(t0, t, x) has a unique solution
x = u(t, tQ, XQ) in V.

(a) By assumption, there is a ball J0 c I of center a and a ball B0 <= H
of center b and radius r such that in J0 x B0, \\f(t, x)\\ ^ M, and ||/(f, xj —
f(t,x2)\\<k'\\xi-x2\\ for teJ0,xt,x2 in B0. By (10.4.5) and (10.5.2)
there is an open ball JA c: J0 of center t0 and a unique solution v of (10.4.1)
defined in J1? taking its values in H and such that v(a) = b. We are going to
see that the open ball V of center b and radius r/2, and the open ball J of center
a and radius p, answer our specifications as soon as p is small enough. Applyin [0, -h oo [. There